A bead of mass M is threaded on the rod of length L. The rod rotates in the horizontal plane with angular velocity (omega). Write the lagrangian and the equation of motion of the bead.
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A bead of mass M is threaded on the rod of length L. The rod rotates in the horizontal plane with
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- You are designing an RPG (role-playing game) for a gaming console and have decided to use an open world design, where players can explore the terrain freely, encountering enemies by chance. 8. Your design team has coded this in-game world to exist on the circle x² + y? < 900 Page 3 on the xy-plane. At any point (x, y) in this world you've also associated a danger function d(x, y) that measures how likely it is to encounter an enemy at that point. Thus high values of d(x,y) correspond to dangerous points, while low values of d(x, y) correspond to safe points. If d(x, y) = e¬a²y, find the safest point(s) and most dangerous point(s) in-game.A block of mass m = 240 kg rests against a spring with a spring constant of k = 550 N/m on an inclined plane which makes an angle of θ degrees with the horizontal. Assume the spring has been compressed a distance d from its neutral position. Refer to the figure. (a) Set your coordinates to have the x-axis along the surface of the plane, with up the plane as positive, and the y-axis normal to the plane, with out of the plane as positive. Enter an expression for the normal force, FN, that the plane exerts on the block (in the y-direction) in terms of defined quantities and g. (b) Denoting the coefficient of static friction by μs, write an expression for the sum of the forces in the x-direction just before the block begins to slide up the inclined plane. Use defined quantities and g in your expression. (c) Assuming the plane is frictionless, what will the angle of the plane be, in degrees, if the spring is compressed by gravity a distance 0.1 m? (d) Assuming θ = 45 degrees and the…Consider the “Foucault pendulum”, as shown below. Foucault set up his 1851 spherical pendulum (of mass m and length L) experiment in the Pantheon dome of Paris, showing that the plane of oscillation rotates and takes about 1.3 days to fully revolve around. This demonstrated the extent to which Earth’s surface is not an inertial reference frame (e.g., role of the Coriolis force). Your task here is to determine (but not solve) the equations of motion.